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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 10, OCTOBER 2015 5643

On Delay Constrained Multicast Capacity ofLarge-Scale Mobile Ad Hoc Networks

Shan Zhou and Lei Ying, Member, IEEE

Abstract— This paper studies the delay constrained multicastcapacity of large-scale mobile ad hoc networks (MANETs).We consider a MANET that consists of ns multicast sessions.Each multicast session has one source and p destinations. Eachsource sends identical information to the p destinations in itsmulticast session, and the information is required to be deliveredto all the p destinations within D time-slots. Assuming thewireless mobiles move according to a 2-D independently and iden-tically distributed mobility model, we first prove that the capacity

per multicast session is O(min{1, (log p)(log(ns p))(D/ns)1/2}).1

We then propose a joint coding/scheduling algorithm achievinga throughput of �(min{1, (D/ns)

1/2}). Our simulation resultssuggest that the same scaling law also holds under random walkand random waypoint models.

Index Terms— Mobile ad hoc networks, multicast,delay-throughput trade-off, scaling law.

I. INTRODUCTION

W IRELESS technology has provided an infrastructure-free and fast-deployable method to establish commu-

nication, and has inspired emerging networks such as mobilead hoc networks (MANETs), which have broad applicationsin personal area networks, emergency/rescue operations, andmilitary battlefield applications. For example, the ZebraNet [1]is an MANET used to monitor and study animal migrationsand inter-species interactions, where each zebra is equippedwith an wireless antenna and pairwise communication isused to transmit data when two zebras are close to eachother. Another example is the mobile-phone mesh networkproposed by TerraNet AB (a Swedish company) [2], where

Manuscript received December 20, 2009; revised December 17, 2010;accepted July 8, 2015. Date of publication July 23, 2015; date of currentversion September 11, 2015. This work was supported in part by DTRA underGrant HDTRA1-08-1-0016 and in part by NSF under Grant CNS-1264012.This paper was presented at the Proceedings of the IEEE INFOCOMMini-Conference in 2010.

S. Zhou was with Iowa State University, Ames, IA 50011 USA, andalso with Arizona State University, Tempe, AZ 85281 USA. He isnow with LinkedIn Corporation, Sunnyvale, CA 94085 USA (e-mail:[email protected]).

L. Ying is with the School of Electrical, Computer and EnergyEngineering, Arizona State University, Tempe, AZ 85281 USA (e-mail:[email protected]).

Communicated by R. Berry, Associate Editor for Communication Networks.Color versions of one or more of the figures in this paper are available

online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2015.24602351Given non-negative functions f (n) and g(n): f (n) = O(g(n)) means

there exist positive constants c and m such that f (n) ≤ cg(n) for alln ≥ m; f (n) = �(g(n)) means there exist positive constants c and msuch that f (n) ≥ cg(n) for all n ≥ m; f (n) = �(g(n)) means thatboth f (n) = �(g(n)) and f (n) = O(g(n)) hold; f (n) = o(g(n))means that limn→∞ f (n)/g(n) = 0; and f (n) = ω(g(n)) means thatlimn→∞ g(n)/ f (n) = 0.

the participated mobile phones form a mesh network and cantalk to each other without using the cell infrastructure.

Over the past few years, there has been a lot of interestin understanding the capacity of MANETs under a rangeof mobility models [3]–[20]. Most of this work assumesunicast traffic flows and studies the unicast capacity. However,multicast flows are expected to be predominant in many ofemerging applications. For example, in battlefield networks,commands need to be broadcast in the network or sent to aspecific group of soldiers. In a wireless video conference, thevideo needs to be sent to all participants. To support theseemerging applications, it is imperative to have a fundamentalunderstanding of the multicast capacity of wireless networks.Shakkottai et al. [21] and Li et al. [22] proved that the mul-ticast capacity of a static ad hoc network is O

(1√

ns log(ns p)

)

per multicast session. Lee et al. [23] investigated the multicastcapacity of delay tolerant networks without delay constraints.In [24], the multicast-capacity and delay tradeoff is establishedfor a specific routing/scheduling algorithm.

In this paper, we study the multicast capacity of large-scaleMANETs under a general delay constraint D. The multicastproblem differs from the unicast problem in the followingaspects:

• Each multicast session has multiple destinations, so theprobability that a packet is within the transmission rangeof its destination(s) is higher than that in the unicastscenario. On the other hand, in the multicast scenario,the information needs to be transmitted reliably fromthe source to all its destinations, which requires moretransmissions than that in the unicast scenario.

• Due to the broadcast nature of wireless communication,all mobiles in the transmission range of a transmittercan simultaneously receive the transmitted packet. In theunicast scenario, only one mobile (the destination ofthe packet) is interested in receiving the packet. In themulticast scenario, all the destinations belonging to thesame multicast sessions are interested in the packet.Thus, one transmission can result in multiple successfuldeliveries in the multicast scenario, which can increasethe capacity of MANETs.

Because of these differences, the multicast capacity ofMANETs is different from the unicast capacity. The focus ofthis paper is to understand the scaling law of delay-constrainedmulticast in MANETs.

The scaling approach is introduced in [25], and has beenintensively used to study the capacity of wireless ad hocnetworks including both static and mobile networks.

0018-9448 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

5644 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 10, OCTOBER 2015

Fig. 1. A MANET with two multicast sessions, where dst(1,1) and dst(1,2)are the destinations of src 1, and dst(2,1) and dst(2,2) are the destinationsof src 2. A mobile can serve as a relay for other multicast sessions.

We consider a MANET consisting of ns multicast sessions.Each multicast session has one source and p destinations.The wireless mobiles are assumed to move according to atwo-dimensional independently and identically distributed(2D-i.i.d) mobility model. Each source sends identical infor-mation to the p destinations in its multicast session, and theinformation is required to be delivered to all the p destinationswithin D time-slots. The main contributions of this paperinclude:

• Given a delay constraint D, we provethat the capacity per multicast session is

O(

min{

1, (log p)(log (ns p))√

Dns

}), when D =

ω(

3√

nslog(ns)2(log(ns p))2

). We then propose a joint

coding-scheduling algorithm achieving a throughput

of �(

min{

1,√

Dns

}). The algorithm is developed

based on an information theoretical approach, where weexploit erasure codes to guarantee reliable multicast. Theidea of exploiting coding has been used in MANETswith unicast flows [15], [16], [18] and mobile sensornetworks [26].

• We evaluate the performance of our algorithm using sim-ulations. We apply the algorithm to the 2D-i.i.d. mobilitymodel, random-walk model and random waypoint model.The simulations confirm that the results obtained from the2D-i.i.d. model hold for more realistic mobility modelsas well.

Finally, we would like to remark that (a) Similar to theunicast scenario [3], the mobility significantly improves thethroughput. While the multicast capacity of a static networkis O

(1√

ns log ns p

), our algorithm achieves a throughput of

�(1) when D = ns . (b) Our result again demonstratesthe substantial benefit of using coding. While the algorithmin [24] achieves a throughput of �

(1

p√

ns p log p

)with an

average delay �(√

ns p log p), our algorithm achieves a much

higher throughput �(

4√

p log pns

)with a hard delay constraint

�(√

ns p log p).

II. MODEL

We consider a mobile ad hoc network with ns multicastsessions as in Figure 1. Each multicast session consists of

Fig. 2. The two transmissions can succeed simultaneously if the distancebetween node j and node k is larger than (1+�)αk and the distance betweennode i and node h is larger than (1 + �)αi .

one source node and p destination as shown in Figure 1.Each node is either a source or a destination. Therefore,there are n � ns(p + 1) mobiles in the network. A sourcesends identical information to all its destinations, and mobilesnot belonging to the multicast session can serve as relays.All mobiles are assumed to be positioned in a unit torus,where the left and right edges are connected, and top andbottom edges are also connected. We further assume themobiles move according to a two-dimensional identical andindependently distributed mobility model (2D-i.i.d. mobilitymodel) [6] such that: (i) at the beginning of each timeslot, a mobile randomly and uniformly selects a point fromthe unit torus and instantaneously moves to that point; and(ii) the positions of mobiles are independent across mobilesand time slots.

Each mobile is equipped with a wireless antenna, and cancommunicate with another mobile within the transmissionradius. We first assume that each mobile can adapt powerand use an arbitrary transmission radius to obtain a gen-eral upper-bound on the delay-constrained multicast capacity.Then we propose a joint coding/scheduling algorithm that(i) achieves a near-optimal throughput, and (ii) requires onlytwo transmission radii {L1, L2}, where L1 is for sending outinformation from sources, and L2 is for delivering packets tothe destinations.

We adopt the protocol model introduced in [27] for thewireless interference. Let αi denote the transmission radius ofnode i, then a transmission from node i to node j is successfulunder the protocol model if and only if the following twoconditions hold: (i) the distance between nodes i and j isless than αi , and (ii) if mobile k is transmitting at the sametime, then the distance between node k and node j is at least(1 + �)αk (see Figure 2), where the � > 0 defines a guardzone around the transmission. Notice that multiple destinationsmay receive the same packet from a single transmission if theyare all in the transmission range of the source and are notinterfered. We adopt this protocol model because the mobilesare allowed to transmit with different powers (i.e., differenttransmission radii) under this model, which allows us toobtain a general upper bound on the multicast capacity ofMANETs. Note that under this protocol model, the receiverof node i associates an exclusion region which is a disk withradius �αi/2 and centered at the receiver of node i. It has beenshown in [25] that exclusion regions associated with successful

ZHOU AND YING: ON DELAY CONSTRAINED MULTICAST CAPACITY OF LARGE-SCALE MANETs 5645

TABLE I

NOTATIONS

transmissions should be disjoint from each other. We furtherassume that each successful transmission can transmit W bitsper time-slot.

A. Delay Constraint

We assume a hard delay constraint D in this paper. A packetis said to be successfully multicast if all p destinations receivethe packet within D time slots after the packet is moved tothe head of the queue of the source node.

B. Multicast Throughput

Let λ denote the multicast throughput per multicast sessionand �si [T ] denote the number of bits that are successfullydelivered to all the destinations of multicast session i up totime T . A multicast throughput λ is said to be supportableunder delay constrain D and loss probability ε if thereexists n0 such that for any n > n0, there exists a coding,routing, and scheduling algorithm such that the average num-ber of bits successfully delivered in each multicast session isgreater than λ w.h.p., i.e., every bit is successfully multicastwith a probability at least 1 − ε, and

limT →∞ Pr

(�si [T ]

T> λ,∀i

)= 1 (1)

Before we continue, we list some important notations used inthe paper. A summary of noation used throughout this papercan be found in Table I.

III. MAIN RESULTS AND INTUITION

In this section, we present the main results of this paperalong with the key intuition. We use the virtual channel ideaproposed in [18] to analyze heuristically our system, wherea virtual channel is a logical representation of the end-to-endtransmission of a packet from its source to its destination.The virtual channels characterize both randomness due to node

Fig. 3. The three phases of a typical delivery.

Fig. 4. The virtual channel representation of a multicast session.

mobility and wireless capacity constraint due to interference.In this virtual channel model, we assume the packets aretransmitted via two-hop transmission. In the first hop, a packetis transmitted from its source to relays around the source.In the second hop, a packet is transmitted from a relay toits destination. We use this two-hop transmission model toexplain the key intuition that leads to the multicast scalinglaw. In the following sections, we will rigorously derive themulticast scaling law without assuming this two-hop transmis-sion scheme.

Under the two-hop transmission scheme, a successfuldelivery consists of three phases (see Figure 3):

• Phase-I, the packet is transmitted from the source to oneor multiple relay nodes;

• Phase-II, a relay moves to the neighborhood of thedestination(s) of the packet; and

• Phase-III, the relay sends the packet to the destination(s).We now model each phase as a separate virtual channel(see Figure 4):

• Reliable Broadcasting Channel: Under the protocolmodel, the exclusion regions of successful transmissionsare disjoint from each other. To simplify our heuristicanalysis, we assume all sources use a common transmis-sion radius L1 for sending out the information. We alsoassume each exclusion region has an area π(L1)

2.2

Here we omit the constant � for simplicity. Thus, the

2Note these two assumptions, along with other assumptions introduced inthis section, are for the purpose of a heuristic argument. Our results holdwithout these assumptions.


number of simultaneous broadcasts at one time slot is atmost 1

π(L1)2 . On average, each source has P1 fraction of

time to broadcast its packet, where

P1 = 1

π(L1)2ns.

The throughput of a broadcasting channel is:

W

π(L1)2ns.

On average, there are π(L1)2n mobiles in a disk with

area π(L1)2. Therefore, each broadcast creates π(L1)

2nduplicate copies in the network.

• Unreliable Relay Channel (Erasure Channel): We assumethat all relays use a common transmission radius L2 forsending packets to their destinations. The probability thata duplicated packets does not fall into the transmissionrange of a specific destination in D consecutive timeslots is

Pmiss = (1 − π(L2)2)D.

Recall that after being sent out from the source, eachpacket has π(L1)

2n copies. So the probability that noneof the duplicated packets falls into the transmission rangesof the specific destination in D consecutive time slots is

Pmiss2 = (1 − π(L2)2)Dπ(L1)

2n,

which is the erasure probability of a relay channel shownin Figure 4.

• Reliable Receiving Channel: Consider the transmissionsfrom relays to destinations. When a packet is broadcastby a relay, all the destinations that are in the transmissionrange of the relay can receive the packet, which resultsin multiple deliveries. We name one of the deliveriesas target delivery, and the rest as free-ride deliveries.Under the protocol model, all exclusion regions asso-ciated with the targeted deliveries should be disjointfrom each other. With a common transmission radius L2,a successful target delivery associates an exclusion regionwith area π(L2)

2. So the number of simultaneous targetdeliveries is no more than

W

π(L2)2 .

Furthermore, along with each target delivery, we have

(p − 1)π(L2)2

free-ride deliveries on average. Thus, we can expect

W (1 + (p − 1)π(L2)2)

π(L2)2

deliveries at each time slot. Recall that a source packetneeds to be delivered to all p destinations, so the through-put per multicast session is

W1 + (p − 1)π(L2)

2

ns pπ(L2)2 = W

ns pπ(L2)2 + W (p − 1)

ns p

bits per time slot.

Based on the virtual channel representation, we canconclude heuristically that

λ = maxL1,L2

min

{(1 −

(1 − π(L2)

2)π D(L1)

2n)

W

π(L1)2ns,

W

ns pπ(L2)2 + W (p − 1)

ns p

}

= �

(√D

ns

),

where the transmission radii L1 and L2 that solvethe maximization problem are L∗

1 = �(

12√ns

)and

L∗2 = �

(1

4√

p2 Dns

).

We would like to comment that all analysis above is heuris-tic, which however captures the key properties determiningthe delay constrained multicast capacity. The rigorous analysiswill be presented in the rest of the paper, where we will provethe following main results:

Main Result 1: Given the delay constraint D, the multicastcapacity λ (per multicast session) is

λ =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, if D = o

(3

√ns

(log p)2(log(ns p))2

);

O(1), if D = ω

(ns


);

O((log p)

(log(ns p))√

Dns

), otherwise.

Main Result 2: There exists a joint coding/scheduling

algorithm that achieves a throughput of �(√

Dns

)when D is

both ω( 3√

ns log(ns p)) and o(ns).

λ =

⎧⎪⎪⎨⎪⎪⎩

�(1), if D = ω

(ns


);

�

(√D

ns

), if D = ω( 3

√ns log(ns p)) and D = o(ns).

IV. UPPER BOUND

In this section, we present an upper-bound on the multicastcapacity of MANETs. Recall that multicast in MANETs isdifferent from unicast in the following aspects:

• A source packet is destined to p destinations, so hasa higher probability being deliverable compared to theunicast case, where a packet is said to be deliverable ifat least one destination is within its transmission range.

• When a packet is transmitted, it can be received by all thedestinations in the transmission range, which increasesthe efficiency of the transmission.

To investigate the upper-bound on multicast throughput λ,we would like to introduce some notations first. We say apacket is successfully delivered to a destination d j if thedestination receives the packet before the deadline expires.We denote by �d j [T ] the number of successfully deliveredbits to destination d j up to time T . Further, let �[T ] denotethe total number of bits successfully delivered up to time T,i.e., �[T ] = ∑

di�di [T ]. Note that a successful multicast


packet is a packet that is successfully delivered to all itsdestinations, so

∑si

�si [T ] ≤ �[T ]/p. Recall that if onepacket is received by multiple destinations in the transmissionrange, one of this multiple successful deliveries is called targetdelivery and the others are called free-ride deliveries. Let B[T ]denote the number of bits delivered by target deliveries up totime T .

Next we derive an upper bound on the multicast throughputby two steps: First, we obtain an upper-bound on B[T ]. Then,we establish the relation between �[T ] and B[T ], which leadsto the upper bound on �[T ].

Given a destination, we classify the successfully deliveredbits into two categories: (i) those bits that are transmitteddirectly from sources to destinations, and (ii) those bits thatare delivered by the relays. In the following lemmas, we boundthe number of bits in each category. We define B S[T ] to bethe number of bits that are directly transmitted from sourceto destinations by target deliveries, up to time T . We furtherdefine B R[T ] to be the number of bits that are transmitted fromrelays to destinations by target deliveries, up to time T . Thefollowing two lemmas establish the upper-bounds on B S[T ]and B R[T ] respectively.

Lemma 1: Assuming the 2D-i.i.d. mobility and the protocolmodels, we have

E[B S[T ]] ≤ W T

√32

�2

√ns p (2)

Proof: The proof is presented in Appendix A. �Lemma 2: Assuming the 2D-i.i.d. mobility and the protocol

models, we have

E[B R[T ]] ≤ W T

√32

�2 (p + 1)√

ns D (3)

Proof: The proof is presented in Appendix B. �Based on Lemmas 1 and 3, we obtain an upper-bound

on B[T ] :

E[B[T ]] = E[B S[T ]] + E[B R[T ]]

≤ W T

√32

�2

((p + 1)

√ns D + √

ns p)

. (4)

Next we investigate the relation between �[T ] and B[T ].Lemma 3: Assuming the 2D-i.i.d. mobility and the protocol

models, we have

E[�[T ]] ≤ 5κ log(ns p)E [B[T ]]+ 16κW T

�2 p(log p) log(ns p).

where κ > 0 is a constant independent of ns and p.Proof: The proof is presented in Appendix C. �

From the definition of the multicast capacity λ, we haveλ ≤ �[T ]/(T ns p) because a successful multicast requiresp successful deliveries. To that end, we can derive the fol-lowing theorem on the delay-constrained multicast capacityof MANETs.

Theorem 4: The delay constrained multicast capacity underthe 2D-i.i.d. mobility and protocol model is

λ =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

0, if D = o

(3√

ns(log p)2(log(ns p))2

);

O(1), if D = ω(


);

O((log p)

(log(ns p))√

Dns

), otherwise.

(5)

Proof: Note when D = o

(3√


), it is

easy to verify that Dλ = o(1). This means under the delayconstraint D, the throughput per D time slots is less thanone bit. We view bit is the smallest quantity for information,so the multicast capacity is zero in this case.

Next when D = ω(


), it can be easily

verified that (log p)(log(ns p))√

Dns

= ω(1). However, eachsource can send out at most W bits per time-slot, so λ ≤ W,which leads to the second case.

For the last case, from inequality (4) and Lemma 3, we canconclude that

E[�[T ]] ≤ 5κ log(ns p)

√32

�2 W T((p + 1)

√ns D + √

ns p)

+16κW T

�2 p(log p) log(ns p)

= O(

T p√

ns D(log p) log(ns p))

Recall that λ ≤ �[T ]/(T ns p), we can then conclude that

λ = O((log p)(log(ns p))

√Dns

). �

V. JOINT CODING-SCHEDULING ALGORITHM

In this section, we propose new algorithms that almostachieve the upper bound obtained in the previous section.We consider two different cases: ns = �(1) and ns = ω(1).For the first case, show that a simple round robin schedulingalgorithm achieves the upper bound. For the second case, weintroduce a joint coding-scheduling algorithm that leverageserasure-codes and yields a throughput that differs from theupper bound by a poly-logarithmic factor.

A. Case 1: ns = �(1)

When ns = �(1), a simple scheme is to let the sourcesbroadcast their packets to all the mobiles in the network in around-robin fashion. It is easy to see that under this simplealgorithm, the throughput per multicast session and the delayare �(1).

B. Case 2: ns = ω(1)

To approach the upper bound obtained in Theorem 4. In thissubsection, we propose a scheme which exploits coding.

In our algorithm, we code data packets into coded packetsusing rate-less codes — Raptor codes [28]. Assume that Qdata packets are coded using the Raptor codes. The receivercan recover the Q data packets with a high probability after it


receives any (1+δ)Q distinct coded packets [28]. It is worthyto point out that any other capacity achieving code for binaryerasure channel also works for our algorithm.

We use a modified two-hop algorithm introduced in [3],which consists of two major phases — broadcasting andreceiving. At the broadcasting phase, we partition the unittorus into square cells (broadcasting cells) with each side oflength equal to 1/

√ns . All sources use a transmission radius√

2/√

ns in the broadcasting phase. To avoid interferencecaused by transmissions in neighboring cells, the cells arescheduled according to the cell scheduling algorithm intro-duced in [25] so that each cell can transmit for a constantfraction of time during each time slot, and concurrent trans-missions do not cause interference. We assume each cell cansupport a transmission of two packets during each time slot.In the receiving step, the unit square is divided into square cells

(receiving cells) with each side of length equal to 1/4√

ns p2 D.

The transmission radius used in this phase is√

2/4√

ns p2 D,

where D = 0.5D is defined for the simplicity of notation inthe following discussion. Note that the two transmission radiiused in this algorithm are derived from the virtual channelpresentation.

Similar as in [18], we define four classes of packets inthe network. We also define and categorize packets into fourdifferent types.

• Data packets: Uncoded data packets.• Coded packets: Packets generated by Raptor codes.• Duplicate packets: A coded packet may be broadcast to

other nodes to generate multiple copies. Those copies arecalled duplicate packets.

• Deliverable packets: Duplicate packets that are within thetransmission range of one of the packet’s destinations.

Joint Coding-Scheduling Algorithm: We group every D = 2Dtime slots into a supertime slot. At each supertime slot, thenodes transmit packets as follows:

(1) Raptor Encoding: Each source takes D500

√D/ns data

packets, and uses Raptor codes to generate D codedpackets.

(2) Broadcasting: This step consists of D time slots. At eachtime slot, in each cell, one source is randomly selectedto broadcast a coded packet to 9(p + 1)/10 mobiles inthe cell (the packet is sent to all mobiles in the cell if thenumber of mobiles in the cell is less than 9(p + 1)/10).

(3) Deletion: After the broadcasting phase, all nodes checkthe duplicate packets they have. If more than oneduplicate packet belong to the same multicast session,randomly keep one and drop the others.

(4) Receiving: This step requires D time slots. At each timeslot, if a cell contains no more than two deliverablepackets, the deliverable packets are broadcast in the cell;otherwise, no node in the cell attempts to transmit. At theend of this step, all undelivered packets are dropped. Thedestinations decode the received coded packets usingRaptor decoding.

Theorem 5: Assume that the delay constraint D = 2D isboth ω( 3

√ns log(ns p)) and o(ns). For a sufficiently large ns ,

at the end of each super time slot, every source successfullymulticast

D

500

√D

ns

packets with a probability 1 − 1ns p .

Proof: We follow the analysis in [18] to prove thefollowing three steps:

Step 1: During the broadcasting step, with a high probabil-ity, a source sends out D

3 coded packets;Step 2: After the deletion step, with a high probability,

a source has at least 2D15 coded packets, each having more

than 4p5 duplicate copies;

Step 3: Each destination receives more than D400

√Dns

distinctcoded packets after the broadcasting, which guarantees that the

destination can decode the original D500

√Dns

data packets witha high probability.

Analysis of Step 1: Let Bi [t] denote the event that node ibroadcasts a coded packet to 9(p + 1)/10 mobiles at timeslot t . According to the definition of Bi [t], we have that

Pr (Bi [t]) = Pr (≥ 9 p/10 mobiles in the cell)

× Pr (i is selected| ≥ 9 p/10 mobiles in the cell)

≥ Pr (≥ 9 p/10 destinations in the cell)

× Pr (i is the only source in the cell) .

Since the nodes are uniformly and randomly positioned,from the Chernoff bound, we have

Pr (≥ 9 p/10 destinations in the cell) ≥ 1 − 2e− p300 .

Note that there are ns sources in the network, so

Pr (Bi [t]) ≥(

1 − 2e− p300

)(1 − 1

ns

)ns−1

,

which implies that for large p and ns , we have

Pr (Bi [t]) ≥ 0.36.

Then from the Chernoff bound again, we have that for suffi-ciently large D,

Pr

⎛⎝

D∑t=1

1Bi [t ] ≥ D

3

⎞⎠ ≥ 1 − e− D

3000 (6)

Thus, with high probability, more than D/3 coded packets arebroadcast, and each broadcast generates 9 p/10 copies.

Analysis of Step 2: Recall that after the deletion, a mobilecontains at most one packet for each multicast session. We nextstudy the number of coded packets that have more than 4 p/5duplicate copies.

Note that the number of duplicate packets belonging tosession i and left in the network after the deletion is equalto the number of distinct mobiles receiving duplicate packetsfrom session i. Assume that source i sends out Db codedpackets. The number of duplicate copies left after the deletionis the same as the number of nonempty bins of the followingballs-and-bins problem: There are ns p − 1 bins. At each


time slot, 9 p/10 bins are selected to receive a ball in each ofthem. This process is repeated by Db times.

Let N1 to be the number of duplicate packets belonging tomulticast session i after the deletion. From [18, Lemma 22],3

we have

Pr (N1 ≥ (1 − δ)(ns p − 1) p1) ≥ 1 − 2e−δ2(ns p−1) p1/3,

where

(ns p − 1) p1 = (ns p − 1)

(1 − e−Db× 9p

10 × 1ns p−1

)

≥ (ns p − 1)

(1 − e− 9Db

10ns

)

≥ (ns p − 1)

(9Db

10ns− 1

2

(9Db

10ns

)2)

≥ 44

49Db p.

where the last inequality holds for sufficiently large ns (recallthat D = o(ns) under the assumption of the theorem).Choosing δ = 1/50, we have that for sufficiently large ns

and p,

Pr

⎛⎝ N1 ≥ 22

25Db p

∣∣∣∣D∑

t=1

1Bi [t ] = Db

⎞⎠ ≥ 1 − 2e− D

10000 . (7)

Given that there are more than 22Db p/25 duplicate packetsleft in the network, we can easily verify that more than2Db/5 coded packets will have 4 p/5 duplicate copies becauseotherwise less than 22Db p/25 duplicate packets would be left.Letting Ai denote the number of coded packets of session i,which has more than 4 p/5 duplicate packets after the deletion,we have

Pr

⎛⎝ Ai ≥ 2D

15

∣∣∣∣∣D∑

t=1

1Bi [t ] ≥ D

3

⎞⎠ ≥ 1 − 2e− D

10000 . (8)

Note that after the deletion, all duplicate packets belongingto the same multicast session are carried by different mobilenodes.

Analysis of Step 3: We consider a coded packet of multicastsession i, which has at least 4p

5 duplicate copies after thedeletion. Let Dl [t] denote the event that the coded packet isdelivered to its l th destination at time slot t .

First we consider the probability that one of the duplicatecopies of the coded packet is in the same cell with itsl th destination. In the receiving phase, we use the cell with

each side of length equal to 1/4√

ns p2 D, so the averagenumber of nodes in each cell is

ns(p + 1)√ns p2 D

≥√

ns

D.

Recall that the duplicate packets belonging to the same mul-ticast session are carried by distinct mobiles after the deletion,so their mobilities are independent. Assuming the number of

3For the reader’s convenience, we provide this lemma and its proof inAppendix E.

duplicate copies of the coded packet under consideration is M,we have

Pr(only one copy is deliverable to the l th destination

)

= M1√

ns p2 D

⎛⎝1 − 1√

ns p2 D

⎞⎠

M−1

.

Note that M < p, so as ns → ∞, we have⎛⎝1 − 1√

ns p2 D

⎞⎠

M−1

→ e− 1√

ns D → 1.

For sufficiently large ns , we have

Pr(only one copy is deliverable to the l th destination

)

≥ 39

50√

ns D. (9)

Next, we consider the probability that the duplicate copy isdelivered given that it is the only copy which is deliverableto the l th destination. Suppose we have M nodes in thecell containing the l th destination. According to the Chernoffbound, we have

Pr

(M ≤ 11

10

√ns

D

)≥ 1 − e

− 1300

√nsD . (10)

Note the deliverable copy to the l th destination will be deliv-ered if the M−2 other mobiles (other than the mobile carryingthe copy and the l th destination for the copy) do not carrydeliverable packets and there are no deliverable packets forthe M − 2 mobiles.

Now given K mobiles already in the cell, we study theprobability that no more deliverable packet appears when weadd another mobile. First, the new mobile should not bethe destination of any duplicate packets already in the cell.Each mobile carries at most D duplicate packets, so at mostK D duplicate packets are already in the cell. Each duplicatepacket has p destinations. For each duplicate packet, we have

Pr (the new mobile is its destination) = p

ns(p + 1) − K.

Thus, from the union bound, we have

Pr (the new mobile is a new destination)

≤ pK D

ns(p + 1) − K. (11)

Note that each source sends out no more than D duplicatepackets and each duplicate packet has no more than p copies,so at most K D p mobiles carry the duplicate packets towardsthe K existing mobiles in the cell, and

Pr (new added mobile brings new deliverable packets)

≤ K D p

ns(p + 1) − K. (12)

From inequalities (11) and (12), we can conclude that theprobability that the new added mobile does not change the


number of deliverable packets in the cell is greater than

1 − 2K D p

ns(p + 1) − K.

Starting from the mobile carrying the duplicate packet and thel th destination of the packet, the probability that the numberof deliverable packets does not change after adding additionalM − 2 mobiles is greater than

M∏K=2

(1 − 2K D p

ns(p + 1) − K

)≥

(1 − 2M D p

ns(p + 1) − M

)M−2

.

When M ≤ 1110

√ns

D, we have that for sufficiently large ns,

2M D p

ns(p + 1) − M

(M − 2

) ≤ 2.5,

andM∏

K=2

(1 − 2 pK D

ns(p + 1) − K

)≥ e−2.5. (13)

Now according to inequalities (9), (10), and (13), we canconclude that for sufficiently large ns ,

Pr (Dl [t]) ≥ 1

16

1√ns D

, (14)

which implies at each time slot, a coded packet with at least4 p/5 duplicate copies is delivered to its l th destination with aprobability at least 1

16√

ns D.

Note at each time slot, one destination can receive at mostone packet. So the number of distinct coded packets deliveredto the l th destination of multicast session i is the same as thenumber of nonempty bins of following balls-and-bins problem:

Suppose we have 2D15 bins and one trash can. At each time slot,

we drop a ball. Each bin receives the ball with probability1

16√

ns D, and the trash can receives the ball with probability

1 − P, where

P = D

120√

ns D.

Repeat this D times, i.e., D balls are dropped. Note thebins represent the distinct coded packets, the balls representsuccessful deliveries, and a ball is dropped in a specific binmeans the corresponding coded packet is delivered to thedestination.

Let Xi,l denote the number of distinct coded packetsdelivered to destination l of session i. Under the conditionthat at least 2D/15 coded packets of session i have morethan 4 p/5 duplicate copies each, Xi,l is the same as thenumber of nonempty bins of the above balls-and-bins problem.Choose δ = 1/6. From [18, Lemma 22] we have

Pr

(Xi,l ≥ 5

6

2D

15

(1 − e

− D

16√

ns D

)∣∣∣∣∣ Ai ≥ 2D

15

)

≥ 1 − 2e− D

810

⎛⎝1−e

− D16

√ns D

⎞⎠.

Using the fact that 1 − e−x ≥ x − x2/2 for any x ≥ 0

Pr

⎛⎝ Xi,l ≥ D

400

√D

ns

∣∣∣∣∣∣Ai ≥ 2D

15

⎞⎠ ≥ 1 − 2e− D

13000

√Dns . (15)

Note that D√

Dns

→ ∞ under the assumption of the theorem

(D = ω 3√

ns log(ns p)).Summary

Combining inequalities (6), (8) and (15), we can concludethat

Pr

⎛⎝Xi,l ≥ D

400

√D

ns

⎞⎠

≥ 1 − e− D3000 − e− D

10000 − 2e− D13000

√Dns .

Furthermore, for sufficiently large ns and p, we also have

Pr

⎛⎝Xi,l ≥ D

400

√D

nsfor all i, l

⎞⎠

≥ 1 − ns p

(e− D

3000 − e− D10000 − 2e− D

13000

√Dns

)

≥ 1 − 1

ns p,

where the last inequality holds under the assumption of thetheorem (D = ω( 3

√ns log(ns p))). Note that a destination can

decode the D500

√Dns

data packets after getting D400

√Dns

coded

packets with a high probability, so the theorem holds. �From the theorem above, we can see that the throughput

per multicast session is

D

500

√D

ns× 1

2D= �

(√D

ns

),

which is within a poly-logarithmic factor of the upperbound.

VI. SIMULATIONS

In this section, we verify our theoretical results using simu-lations. We implemented the joint coding-scheduling algorithmfor different mobility models, including 2D-i.i.d. mobility,random walk model and random waypoint model. We considera MANET consisting of ns multicast sessions, and the mobilesare deployed in a unit square with ns sub-squares. The randomwalk model and random waypoint model are defined in thefollowing:

• Random Walk Model: At the beginning of each timeslot, a mobile moves from its current sub-square cellto one of its eight neighboring sub-squares or stays atthe current sub-square. Each of the actions occurs withprobability 1/9.

• Random Waypoint Model [29]: At the beginning of eachtime slot, a mobile generates a two-dimensional vectorV = [Vx , Vy], where the values of Vx and Vy areuniformly selected from [1/

√ns , 3/

√ns]. The mobile


Fig. 5. Throughput per multicast session per D time slots with different n′s s.

moves a distance of Vx along the horizontal direction,and a distance of Vy along the vertical direction.

A. Multicast Throughput With Different Numbers of Sessions

In this simulation, the number of multicast sessions (ns )varies from 200 to 1000, each multicast session containsp = 10 destinations, and the delay constraint is set to beD = 200 time slots. Figure 5 shows the throughput per Dtime slots of the three mobility models with different valuesof ns .

4

Our theoretical analysis indicates that the throughput per D

time slots is �(

D√

Dns

). To verify this scaling law, we plot

α(

D√

Dns

)in Figure 5 with α = 0.09. Our simulation result

shows that the throughput per D time slots scales as D√

D/ns

under all three mobility models. Further, the 2D-i.i.d. mobilityhas the largest throughput and the random walk model has thesmallest throughput. This is because the distance a mobile canmove within one time slot is the largest under the 2D-i.i.d.model and is the smallest under the random walk model. Ourresult indicates that the throughput is an increasing function ofthe speed (the distance a mobile can move within a time slot).

B. Multicast Throughput With Different Delay Constraints

In this simulation, we fixed ns = 500 and p = 10,and varied D from 200 to 800 with a step size of 100.We also compared the throughput with function α

√D/ns with

α = 0.075. For all three mobility models, as shown inFigure 6, the simulations confirm that the throughput scalesas

√D/ns .

C. Multicast Throughput With Different Session Sizes

In this simulation, we had ns = 500, the delay constraintis D = 200, and p varies from 4 to 40 with a step size of 4.Figure 7 shows that the throughput is almost invariant withrespect to p.

4In our simulations, we only count the number of distinct packets deliveredthat are successfully delivered before their deadlines expire. We do notconsider coding and decoding in our simulations.

Fig. 6. Throughput per multicast session per D time slots with differentdelay constraints.

Fig. 7. Throughput per multicast session per D time slots with different p′s.

From the simulations above, we can see that the �(√

Dns

)

throughput is achievable not only under 2D-i.i.d. model, butunder more realistic models such as random walk model andrandom waypoint model.

VII. CONCLUSION

In this paper, we studied the delay constrainedmulticast capacity of large-scale MANETs. We first provedthat the upper-bound on throughput per multicast session is

O(

min{

1, (log p)(log (ns p))√

Dns

}), and then proposed a

joint coding-scheduling algorithm that achieves a throughput

of �(

min{

1,√

Dns

}). We also validated our theoretical

results using simulations, which indicated that the resultsbased on the 2D-i.i.d. model are also valid for randomwalk model and random way point model. In our futureresearch, we will study (i) the impact of mobile velocityon the communication delay and multicast throughput; and(ii) the delay constrained multicast capacity of MANETswith heterogeneous multicast sessions, e.g., differentmulticast sessions have different sizes and different delayconstraints.


APPENDIX APROOF OF LEMMA 1

First, we present some important inequalities that will beused to obtain the upper-bound on throughput. Let R[T ]denote the number of bits that are carried by the mobilesother than their sources at time T (including those whosedeadlines have expired), and αB the transmission radius usedto deliver bit B. The following lemma is presented in [18].Inequality (16) holds since the total number of bits transmittedor received in T time slots cannot exceed ns pW T . Inequal-ity (17) holds since the total number of bits transmitted to relaynodes cannot exceed ns(p+1)W T . Inequality (18) holds sinceeach successful target delivery associates an exclusion regionwhich is a disk with radius �αB/2.

Lemma 6: Under the simplified protocol model, thefollowing inequalities hold:

�[T ] ≤ ns pW T (16)

R[T ] ≤ ns(p + 1)W T (17)B[T ]∑B=1

�2

4(αB)2 ≤ W T

π(18)

Let si denote the source of multicast session i, di, j denotethe j th destination of multicast session i, and D(si , t) thedistance between source si and its nearest destination, i.e.,

D(si , t) = min1≤ j≤p

dist(si , di, j )(t).

Thus, we have

Pr (D(si , t) ≤ L) ≤ 1 − (1 − π L2)p ≤ pπ L2,

which implies

E

[T∑

t=1

ns∑i=1

1D(si ,t)≤L

]≤ T ns pπ L2.

Since at most W bits a source can send during each transmis-sion, we further have

E[

B S[T ]]

= E

⎡⎣

B S[T ]∑B=1

1αB≤L

⎤⎦ + E

⎡⎣

B S[T ]∑B=1

1αB>L

⎤⎦

≤ W E

[T∑

t=1

ns∑i=1

1D(si,t)≤L

]+ E

⎡⎣

B S[T ]∑B=1

1αB>L

⎤⎦

≤ W T ns pπ L2 + E

⎡⎣

B S[T ]∑B=1

1αB>L

⎤⎦.

Next, applying the Cauchy-Schwarz inequality to inequal-ity (18), we can obtain that

⎛⎝

B S[T ]∑B=1

αB

⎞⎠

2

≤⎛⎝

B S[T ]∑B=1

1

⎞⎠

⎛⎝

B S[T ]∑B=1

(αB)2

⎞⎠

≤ B S[T ]4W T

π�2 ,

which implies that√

4W T

π�2

√E[B S[T ]] ≥ E

[√4W T

π�2 B S[T ]]

≥ E

⎡⎣

B S[T ]∑B=1

αB

⎤⎦

≥ L E

⎡⎣

B S[T ]∑B=1

1αB>L

⎤⎦

≥ L(

E[B S[T ]] − W T ns pπ L2)

,

where the first inequality follows from the Jensen’s inequality.

Now we choose L =√

E[B S[T ]]2W T ns pπ , we can obtain that

E[B S[T ]] ≤ W T

√32

�2

√ns p.

APPENDIX BPROOF OF LEMMA 2

Denote by H (b) the minimum distance between the relaycarrying bit b and any of the p destinations of the bit duringD consecutive time slots. We have

Pr (H (b) ≤ L) ≤ 1 − (1 − π L2)Dp ≤ π L2 Dp.

Note that inequality (17) shows R[T ] ≤ ns(p +1)W T, whichimplies

E

⎡⎣ ∑

b∈R[T ]1H(b)≤L

⎤⎦ ≤ ns(p + 1)W T π L2 Dp,

and

E[

B R[T ]]

= E

⎡⎣

B R [T ]∑

B R=1

1αSB≤L

⎤⎦ + E

⎡⎣

B R [T ]∑B=1

1αB>L

⎤⎦

≤ E

⎡⎣

B R [T ]∑B=1

1H(B)≤L

⎤⎦ + E

⎡⎣

B R [T ]∑B=1

1αB>L

⎤⎦

≤ ns(p + 1)W T π L2 Dp + E

⎡⎣

B R [T ]∑B=1

1αB>L

⎤⎦.

(19)

Next, applying the Cauchy-Schwarz inequality to inequal-ity (18), we can obtain that

⎛⎝

B R[T ]∑B=1

αB

⎞⎠

2

≤⎛⎝

B R[T ]∑B=1

1

⎞⎠

⎛⎝

B R[T ]∑B=1

(αB)2

⎞⎠

≤ B R[T ]4W T

π�2 ,


which implies that√

4W T

π�2

√E[B R[T ]]

≥ E

[√4W T

π�2 B R[T ]]

≥ E

⎡⎣

B R [T ]∑B=1

αB

⎤⎦

≥ L E

⎡⎣

B R[T ]∑B=1

1αB>L

⎤⎦

≥ L(

E[B R[T ]] − ns(p + 1)W T π L2 Dp),

where the first inequality follows from the Jensen’s inequalityand the last inequality follows from inequality (19).

Since the inequality holds for any L > 0. By choosing

L =√

E[B R [T ]]2W Tπns (p+1)pD , we can obtain that

E[B R[T ]] ≤√

32

�2 W T (p + 1)√

ns D.

APPENDIX CPROOF OF LEMMA 3

We first show that the number of occasions that morethan κ(1 + pγ 2) log(ns p) destinations belonging to the samesessions are in a disk with radius γ is small in Lemma 7.For a destination j, we let H ( j, γ , t) denote the numberof destinations that belong to the same multicast session asnode j and are within a distance of γ from j at time t .We further define

Zγ,κ[T ] =T∑

t=1

∑j

1H( j,γ ,t)≥κ(1+pγ 2) log(ns p).

Lemma 7: There exists κ > 0, independent of ns and p,such that for any γ ∈ (0, 1]

Pr(

H ( j, γ , t) > κ(1 + pγ 2) log(ns p))

≤ e−κ log(ns p),

(20)

and

E[Zγ,κ[T ]] ≤ T

(ns p)2 (21)

hold.Proof: The proof is presented in Appendix D. �

We index the target deliveries using B. Let βB denote thenumber of deliveries associated with target delivery B. Givena γ ∈ [0, 1], we classify the target-deliveries according to αB .We say a target-delivery belonging to class (γ, m) if 2m−1γ ≤αB < 2mγ. Thus, �[T ] can be written as

E[�[T ]] ≤ E

[B[T ]∑B=1

βB1αB<γ

]

+�− log2 γ �∑

m=1

E

[B[T ]∑B=1

βB12m−1γ≤αB<2mγ

].

Fig. 8. H (dB, 2γ, t) ≥ βB .

Note that βB > κ(1 + 4 pγ 2) log(ns p) implies that

H (dB, 2γ, t) ≥ κ(1 + 4 pγ 2) log(ns p)

as shown in Figure 8, where dB is the destination receivingthe target delivery, so we have

βB1αB<γ ≤ κ(1 + 4 pγ 2) log(ns p)

×1 αB<γ

H(dB ,2γ,t)<κ(1+4pγ 2) log(ns p)

+ βB1 αB<γH(dB ,2γ,t)≥κ(1+4pγ 2) log(ns p)

. (22)

Furthermore, it can be easily verified that

E

[B[T ]∑B=1

βB1 αB<γ

H(dB,2γ,t)≥κ(1+4pγ 2) log(ns p)

]+

�− log2 γ �∑m=1

×E

[B[T ]∑B=1

βB1 2m−1γ≤αB<2mγ

H(dB ,2m+1γ,t)≥κ(1+pγ 222m+2) log(ns p)

]

≤(a)pW T

n2s p2

= W T

n2s p

, (23)

where inequality (a) yields from inequality (20).Now from the inequalities (22) and (23), we have for any

0 < γ < 1,

E[�[T ]]

≤ W T

n2s p

+ κ log(ns p)

((1 + 4 pγ 2

)E

[B[T ]∑B=1

1αB<γ

]

+�− log2 γ �∑

m=1

(1 + p22m+2γ 2

)E

[B[T ]∑B=1

12m−1γ≤αB<2mγ

]⎞⎠

≤(b)W T

n2s p

+ κ log(ns p)(

1 + 4 pγ 2)

E [B[T ]]

+ κ log(ns p)

�− logγ / log 2�∑m=1

p22m+2γ 2 W T

π 22m−2�2γ 2

4

≤ W T

n2s p

+ κ log(ns p)(

1 + 4 pγ 2)

E [B[T ]]

+ 64κW T

π�2 log 2(− log γ )p log(ns p)

≤ 5κ log(ns p)E [B[T ]] + 16κW T

�2 p(log p) log(ns p),

(24)


where inequality (b) yields from inequality (18), and the lastinequality holds when γ = 1√

p .

APPENDIX DPROOF OF LEMMA 7

Recall that each multicast session contains p destinations.The probability that a mobile is within a distance of γ fromnode j is πγ 2. Thus, H ( j, γ , t) is a binomial random variablewith p − 1 trials and probability of a success πγ 2, and

E[H ( j, γ , t)] = (p − 1)πγ 2.

Now choose κ such that

κ(1 + pγ 2) log(ns p) > (p − 1)πγ 2. (25)

Note that (p−1)πγ 2

(1+pγ 2) log(ns p)< π for ns p > 3, so we can choose

κ independent of ns and p. Next, define

δ = κ(1 + pγ 2) log(ns p)

(p − 1)πγ 2 − 1,

which is positive due to inequality (25).According to the Chernoff bound [30], we have

Pr(

H ( j, γ , t) > κ(1 + pγ 2) log(ns p))

≤(

eδ

(1 + δ)1+δ

)(p−1)πγ 2

≤(

e

1 + δ

)(p−1)πγ 2(1+δ)

=(

e

1 + δ

)κ(1+pγ 2) log(ns p)

=⎛⎝ e

κ(1+pγ 2) log(ns p)(p−1)πγ 2

⎞⎠

κ(1+pγ 2) log(ns p)

≤(a) e−κ(1+pγ 2) log(ns p)

≤ e−κ log(ns p), (26)

where inequality (a) holds for any κ such that

eκ(1+pγ 2)(log p+log ns )

(p−1)πγ 2

≤ e−1.

Thus, we can conclude that there exists κ > 0, which isindependent of ns and p, such that

E[Zγ,κ[T ]] ≤ E

⎡⎣ ∑

j : j is a destination

T∑t=1

1H( j,γ ,t)≥κγ 2 log(ns p)

⎤⎦

=T∑

t=1

∑j

E[1H( j,γ ,t)≥κγ 2 log(ns p)

]

≤ ns pT e−κ log(ns p),

and the theorem holds by guaranteeing κ > 2.

APPENDIX EBALLS-AND-BINS PROBLEM

The proof of the following lemma about the balls-and-binsproblem is given in [18]. We present the proof here for thereader’s convenience.

Lemma 8: Suppose n balls are independently dropped intom bins and one trash can. After a ball is dropped, theprobability in the trash can is 1 − p, and the probability in aspecific bin is p/m. Using N2 to denote the number of binscontaining at least 1 ball, the following inequality holds forsufficiently large n.

Pr (N2 ≤ (1 − δ)m p2) ≤ 2e−δ2m p2/3; (27)

where p2 = 1 − e− npm .

Proof: Let κi denote the number of balls in bin i. Nextdefine n to be a Poisson random variable with mean n.We consider the case such that n balls are independentlydropped in m bins. Using κi to be number of balls in bin i inthis case, it is easy to see that {κi } are i.i.d. Poisson randomvariables with mean np

m . So we have

Pr (N2 ≤ (1 − δ)m p2)

= Pr

(m∑

i=1

1κi≥1 ≤ (1 − δ)m p2

)

= Pr

(m∑

i=1

1κi≥1 ≤ (1 − δ)m p2

∣∣∣∣∣ n ≥ n

)

≤ Pr(∑m

i=1 1κi≥1 ≤ (1 − δ)m p2)

Pr(n ≥ n).

Since

Pr(1κi≥1 = 1

) = Pr (κi ≥ 1) = 1 − e− npm = p2,

by applying the Chernoff bound [30], we have

Pr

(m∑

i=1

1κi≥1 ≤ (1 − δ)m p2

)≤ e−δ2m p2/3.

which implies for sufficiently large n,

Pr (N2 ≤ (1 − δ)m p2) ≤ √3πne−δ2m p2/3

≤ 2e−δ2m p2/3.

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Shan Zhou received the B.E. degree in automation from Tsinghua University,Beijing, China, in 2006. He received the Ph.D. degree in electrical engineeringfrom Arizona State University, Tempe, US, in 2013. His research interestsinclude distributed scheduling algorithms and power control algorithms inwireless networks.

Lei Ying (M’08) received his B.E. degree from Tsinghua University, Beijing,China, and his M.S. and Ph.D in Electrical and Computer Engineering fromthe University of Illinois at Urbana–Champaign. He currently is an AssociateProfessor at the School of Electrical, Computer and Energy Engineeringat Arizona State University, and an Associate Editor of the IEEE/ACMTRANSACTIONS ON NETWORKING.

His research interest is broadly in the area of stochastic networks, includ-ing cloud computing, communication networks and social networks. He iscoauthor with R. Srikant of the bookCommunication Networks: An Opti-mization, Control and Stochastic Networks Perspective, Cambridge UniversityPress, 2014. The book has been selected as a notable book in the ComputingReviews’ 19th Annual Best of Computing list.

He won the Young Investigator Award from the Defense Threat ReductionAgency (DTRA) in 2009 and NSF CAREER Award in 2010. He wasthe Northrop Grumman Assistant Professor in the Department of Electricaland Computer Engineering at Iowa State University from 2010 to 2012.He received the best paper award at IEEE INFOCOM 2015.

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